Weight Function Approach Research Articles

Building a Sustainable GARCH Model to Forecast Rubber Price: Modified Huber Weighting Function Approach

The unstable and uncertain nature of natural rubber prices makes them highly volatile and prone to outliers, which can have a significant impact on both modeling and forecasting. To tackle this issue, the author recommends a hybrid model that combines the autoregressive (AR) and Generalized Autoregressive Conditional Heteroscedasticity (GARCH) models. The model utilizes the Huber weighting function to ensure the forecast value of rubber prices remains sustainable even in the presence of outliers. The study aims to develop a sustainable model and forecast daily prices for a 12-day period by analyzing 2683 daily price data from Standard Malaysian Rubber Grade 20 (SMR 20) in Malaysia. The analysis incorporates two dispersion measurements (IQR/3 and Sn) and three levels of IO contamination 0%, 10%, and 20%. The results indicate that using the Huber weighting function with the IQR/3 measurement to build the AR(1)-GARCH(2,1) model leads to better sustainability. These findings have the potential to enhance the GARCH model by modifying the weighting function of the M-estimator

Stress intensity factor by combined perturbation method and universal weight functions

The stress intensity factors (SIF) of three-dimensional cracks are known exactly only for a very limited number of shapes and loading scenarios. In this work, the perturbation method is combined to the universal weight function approach to estimate the mode I, stress intensity factor of cracks with crack fronts obtained as perturbations of any known, reference crack shape. This work is limited to cracks and loading scenarios with two planes of symmetry for which the universal weight function could be established. The novel approach is illustrated for a quasi-square crack and long, flexural fatigue cracks in brittle plates. The accuracy and limitations of the proposed method are also discussed. The proposed approach provides the means to approximate the SIF for a variety of shapes and loads without relying on FEM. Furthermore, the proposed approximation is the only method that can accurately approximate the SIF’s for cracks with large aspect ratios.

Fatigue strength assessment of welded joints in the marine environment

Welding is a commonly applied joining method in many applications in arctic and marine conditions, e.g., in ship and offshore structures, and energy production equipment. Such applications are usually subjected to fluctuating load conditions, and during a decades-long service, they may experience millions of load cycles. Consequently, fatigue strength design and acceptable flaw sizes in the welded details of these structures are among the most important design criteria. Multiple fatigue strength assessment approaches exist for assessing the fatigue strength of a welded detail. The present study introduces a numerical and analytical fatigue strength assessment, conducted on a non-load-carrying X-joint, which is a representative joint type used in many steel constructions. Fatigue analyses are carried out following the DNVGL-RP-C203 and BS7910:2013 fatigue design guidelines for offshore steel structures. The stress intensity factors (SIFs) for linear elastic fracture mechanics (LEFM) analyses were obtained using three different methods: the weight function approach, the analytical equations provided in the IIW Recommendations, and by conducting numerical crack propagation analysis using the Franc2D software. All three methods had a good agreement particularly for short crack depths, indicating the applicability of the analytical approaches for the fatigue analyses. The results showed that the consideration of degree of bending at the welded detail is crucial due to the distinguishing notch stress factors of membrane and bending loading, and different stress distributions in the through-thickness direction. In addition, it was found that the LEFM-based fatigue life assessments are significantly more conservative than the life predictions obtained using the structural hot-spot and effective notch stress approaches.

A Family of Higher Order Scheme for Multiple Roots

We have developed a two-point iterative scheme for multiple roots that achieves fifth order convergence by using two function evaluations and two derivative evaluations each iteration. Weight function approach is utilized to frame the scheme. The weight function named as R(υt) is used, which is a function of υt, and υt is a function of ωt, i.e., υt=ωt1+aωt, where a is a real number and ωt=g(yt)g(xt)1m˜ is a multi-valued function. The consistency of the newly generated methods is ensured numerically and through the basins of attraction. Four complex functions are considered to compare the new methods with existing schemes via basins of attraction, and all provided basins of attraction possess reflection symmetry. Further, five numerical examples are used to verify the theoretical results and to contrast the presented schemes with some recognized schemes of fifth order. The results obtained have proved that the new schemes are better than the existing schemes of the same nature.

An optimal eighth order derivative free multiple root finding scheme and its dynamics

<abstract><p>The problem of solving a nonlinear equation is considered to be one of the significant domain. Motivated by the requirement to achieve more optimal derivative-free schemes, we present an eighth-order optimal derivative-free method to find multiple zeros of the nonlinear equation by weight function approach in this paper. This family of methods requires four functional evaluations. The technique is based on a three-step method including the first step as a Traub-Steffensen iteration and the next two as Traub-Steffensen-like iterations. Our proposed scheme is optimal in the sense of Kung-Traub conjecture. The applicability of the proposed schemes is shown by using different nonlinear functions that verify the robust convergence behavior. Convergence of the presented family of methods is demonstrated through the graphical regions by drawing basins of attraction.</p></abstract>

An Optimal Iterative Technique for Multiple Root Finder of Nonlinear Problems

In this paper, an optimal higher-order iterative technique to approximate the multiple roots of a nonlinear equation has been presented. The proposed technique has special properties: a two-point method that does not involve any derivatives, has an optimal convergence of fourth-order, is cost-effective, is more stable, and has better numerical results. In addition to this, we adopt the weight function approach at both substeps (which provide us with a more general form of two-point methods). Firstly, the convergence order is studied for multiplicity m=2,3 by Taylor’s series expansion and then general convergence for m≥4 is proved. We have demonstrated the applicability of our methods to six numerical problems. Out of them: the first one is the well-known Van der Waals ideal gas problem, the second one is used to study the blood rheology model, the third one is chosen from the linear algebra (namely, eigenvalue), and the remaining three are academic problems. We concluded on the basis of obtained CPU timing, computational order of convergence, and absolute errors between two consecutive iterations for which our methods illustrate better results as compared to earlier studies.

Weight function procedure for reduced order fracture analysis of arbitrary flaws in cylindrical pressure vessels

Fracture mechanics calculations using the finite element method can be computationally expensive, which makes them challenging to use in engineering evaluations of flaws in pressure vessels. The weight function (WF) technique is a reduced order fracture modeling method that is widely used to greatly reduce these computational costs. Its computational efficiency is essential for use in probabilistic fracture mechanics evaluations of embrittled nuclear reactor pressure vessels (RPVs), due to the large number of sampled flaws that must be evaluated. Although the WF technique is general, it is typically used only for axis-aligned flaws. Recent discoveries of off-axis flaws in operating nuclear reactors have necessitated detailed simulations of such flaws and the interactions between neighboring flaws. This paper presents a generalized WF approach applicable for analyzing arbitrary flaw geometries in thick-walled cylindrical vessels, including surface-breaking and subsurface flaws, which can either axis-aligned or off-axis, and can account for interactions with other flaws. This approach is demonstrated on representative simulations of multiple flaw geometries in a RPV subjected to transient loading conditions. In all cases, the WF technique gives good comparison with benchmark results from direct simulation with greatly reduced computational effort.

Derivative-Free Iterative Schemes for Multiple Roots of Nonlinear Functions

The construction of derivative-free iterative methods for approximating multiple roots of a nonlinear equation is a relatively new line of research. This paper presents a novel family of one-parameter second-order techniques. Our schemes are free from derivatives and have been designed to find multiple roots (m≥2). The new techniques involve the weight function approach. The convergence analysis for the new family is presented in the main theorem. In addition, some special cases of the new class are discussed. We also illustrate the applicability of our methods on van der Waals, Planck’s radiation, root clustering, and eigenvalue problems. We also contrast them with the known methods. Finally, the dynamical study of iterative schemes also provides a good overview of their stability.

Nearly constant Q models of the generalized standard linear solid type and the corresponding wave equations

Time-domain seismic forward and inverse modeling for a dissipative medium is a vital research topic for investigating the attenuation structure of the earth. Constant [Formula: see text], also called the frequency independence of the quality factor, is a common assumption for seismic [Formula: see text] inversion. We have developed first- and second-order nearly constant [Formula: see text] dissipative models of the generalized standard linear solid type, using a novel [Formula: see text]-independent weighting function approach. The two new models, which originate from the Kolsky model (a nearly constant [Formula: see text] model) and the Kjartansson model (an exactly constant [Formula: see text] model), result in the corresponding wave equations in differential form. Even for extremely strong attenuation (e.g., [Formula: see text]), the quality factor and phase velocity for the two new models are close to those for the Kolsky and Kjartansson models, in a frequency range of interest. The wave equations for the two new models explicitly involve a specified [Formula: see text] parameter and have compact and simple forms. We provide a novel perspective on how to build a nearly constant [Formula: see text] dissipative model, which is beneficial for time-domain large-scale wavefield forward and inverse modeling. This perspective could also help obtain other dissipative models with similar advantages. We also discuss the extension beyond viscoacousticity and other related issues, for example, extending the two new models to viscoelastic anisotropy.

Precise residual stress profile in ion-exchanged silicate glass by modified contour method

In this work, the contour method is modified and used to precisely estimate the residual stress profiles in ion-exchanged aluminosilicate glasses, including the effects of non-homogeneous material properties. The method requires the precise measurements of the fracture surface’s topography and exploits fractographic features to mate and accurately determine the out-of-plane deformation after fracture. Residual stresses are computed from the out-of-plane deformation through a weight function approach rather than by numerical methods, as weight functions can easily account for concentration-dependent material properties. The stress profiles in aluminosilicate glass with three different molten salt soaking times were calculated and compared with the approximate profiles measured by optical waveguide. It was found that the optical waveguide’s estimations under-predicts the surface stress by as much as 10–30 %, likely due to the assumption that the stress optic coefficient is unaffected by the K+↔Na+ exchange.

Fractional q-deformed chaotic maps: A weight function approach.

The fractional derivative holds long-time memory effects or non-locality. It successfully depicts the dynamical systems with long-range interactions. However, it becomes challenging to investigate chaos in the deformed fractional discrete-time systems. This study turns to fractional quantum calculus on the time scale and reports chaos in fractional q-deformed maps. The discrete memory kernels are used, and a weight function approach is proposed for fractional modeling. Rich q-deformed dynamics are demonstrated, which shows the methodology's efficiency.

Crack extension resistance of concrete at low temperatures

The process of determining crack extension resistance curves (KR curves) associated with the cohesive stress distribution for the complete fracture process of a three-point bending concrete beam at low temperatures from 20°C to −40°C is presented in this paper. A total of 32 concrete beams were divided into two groups, based on different ages, 28 d and 120 d. The KR curves were calculated from curves of load against crack mouth opening displacement obtained from test results of the concrete beams at low temperatures by considering the variation of the fracture process zone length. The weight function approach was used to replace numerical integration techniques to improve the computational efficiency. The calculation results show that the KR curves became higher with decreasing temperature and that the fracture process zone length had a significant influence on the KR curves. The stability of the KR curve and the influence of temperature on the fracture process zone length were also analysed.

Modeling of electromechanical response and fracture resistance of multilayer piezoelectric energy harvester with residual stresses

The article focuses on a modeling and subsequent optimization of a novel layered architecture of the vibration piezoceramic energy harvester composed of ZrO2/Al2O3/BaTiO3 layers and containing thermal residual stresses. The developed analytical/numerical model allows to determine the complete electromechanical response and the apparent fracture toughness of the multilayer vibration energy harvester, upon consideration of thermal residual stresses and time-harmonic kinematic excitation. The derived model uses the Euler–Bernoulli beam theory, Hamilton’s variational principle, and a classical laminate theory to determine the first natural frequency, steady-state electromechanical response of the beam upon harmonic vibrations, and also the mechanical stresses within particular layers of the harvester. The laminate apparent fracture toughness is computed by means of the weight function approach. A crucial point is the further optimization of the layered architecture from both the electromechanical response and the fracture resistance point of view. Maximal allowable excitation acceleration of the harvester upon which the piezoelectric layer will not fail is determined. It makes possible to better use the harvester’s capabilities in a given application and simultaneously guarantee its safe operation. Outputs of the derived analytical model were validated with finite element method simulations and available experimental results, and a good agreement between all approaches was obtained.

Efficient optimal families of higher-order iterative methods with local convergence

The main aim of this manuscript is to propose two new schemes having three and four substeps of order eight and sixteen, respectively. Both families are optimal in the sense to Kung-Traub conjecture. The derivation of them are based on the weight function approach. In addition, theoretical and computational properties are fully investigated along with two main theorems describing the order of convergence. Further, we also provide the local convergence of them in Banach space setting under weak conditions. From the numerical experiments, we find that they perform better than the existing ones when we checked the performance of them on a concrete variety of non-linear scalar equations. Finally, we analyze the complex dynamical behavior of them which also provide a great extent to this.

A 4th-Order Optimal Extension of Ostrowski’s Method for Multiple Zeros of Univariate Nonlinear Functions

We present a new optimal class of Ostrowski’s method for obtaining multiple zeros of univariate nonlinear functions. Several researchers tried to construct an optimal family of Ostrowski’s method for multiple zeros, but they did not have success in this direction. The new strategy adopts a weight function approach. The design structure of new families of Ostrowski’s technique is simpler than the existing classical families of the same order for multiple zeros. The classical Ostrowski’s method of fourth-order can obtain a particular form for the simple root. Their efficiency is checked on a good number of relevant numerical examples. These results demonstrate the performance of our methods. We find that the new methods are just as competent as other existing robust techniques available in the literature.

An Optimal Eighth-Order Family of Iterative Methods for Multiple Roots

In this paper, we introduce a new family of efficient and optimal iterative methods for finding multiple roots of nonlinear equations with known multiplicity ( m ≥ 1 ) . We use the weight function approach involving one and two parameters to develop the new family. A comprehensive convergence analysis is studied to demonstrate the optimal eighth-order convergence of the suggested scheme. Finally, numerical and dynamical tests are presented, which validates the theoretical results formulated in this paper and illustrates that the suggested family is efficient among the domain of multiple root finding methods.

Analysis of two-dimensional combustion waves arising in the presence of a competitive endothermic reaction

We consider a system of reaction-diffusion equations describing combustion dynamics. The reaction is assumed to undergo two competitive reactions, one which is exothermic and one which is endothermic. The one-dimensional model has been shown to exhibit complex behaviour, from propagating combustion waves with a constant speed to period doubling cascades and the possibility of chaotic wave speeds. In this study, we extend the combustion model from one to two dimensions by exploring a model of an insulated strip with no heat loss and axially symmetric spread. In particular, we compare and contrast the behaviour of the systems in one and two dimensions. 
 
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The microdosimetric extension in TOPAS: development and comparison with published data

Microdosimetric energy depositions have been suggested as a key variable for the modeling of the relative biological effectiveness (RBE) in proton and ion radiation therapy. However, microdosimetry has been underutilized in radiation therapy. Recent advances in detector technology allow the design of new mico- and nano-dosimeters. At the same time Monte Carlo (MC) simulations have become more widely used in radiation therapy. In order to address the growing interest in the field, a microdosimetric extension was developed in TOPAS. The extension provides users with the functionality to simulate microdosimetric spectra as well as the contribution of secondary particles to the spectra, calculate microdosimetric parameters, and determine RBE with a biological weighting function approach or with the microdosimetric kinetic (MK) model. Simulations were conducted with the extension and the results were compared with published experimental data and other simulation results for three types of microdosimeters, a spherical tissue equivalent proportional counter (TEPC), a cylindrical TEPC and a solid state microdosimeter. The corresponding microdosimetric spectra obtained with TOPAS from the plateau region to the distal tail of the Bragg curve generally show good agreement with the published data.

Watershed initial conditions for propagating waves in a system with bistability: A weight-function approach

We consider non-adiabatic combustion waves arising from a two-step exothermic system. Our previous work showed that in certain parameter regions, the combustion wave can evolve to the “fast” solution branch, the “slow” solution branch or diffuse to the ambient temperature (extinction wave). Here, we are interested to find critical initial temperature profiles which evolve to these three types of steady solutions. For a particular family of temperature profiles, we construct a weight function which can be used to predict which of these three types of waves an initial temperature profile will evolve to.

Quantifying fatigue overload retardation mechanisms by energy dispersive X-ray diffraction

The fatigue crack retardation mechanisms operating after an overload event are investigated for a bainitic steel using high spatial resolution energy dispersive synchrotron X-ray diffraction. The elastic crack-tip strain fields are mapped at mid-thickness of compact tension samples at R-ratios of 0.1 and 0.4. The same overload stress intensity factor (KOL = 60 MPa m1/2) is applied in each case with the cracks then propagating under the same applied stress intensity range, ΔKapp = 27 MPa m1/2. The competing retardation mechanisms are directly quantified and separated, with the associated fatigue crack growth (FCG) rates then being predicted according to a 2-parameter Walker-type assessment and validated against those measured. The stress intensity factor associated with the overload residual stress field is calculated using a weight function approach. For R=0.1, shielding from residual stress controls retardation when crack growth through the overload plastic zone, rpOL, is small (specifically <0.6rpOL). For more extensive crack growth, discontinuous crack closure controls the retardation behaviour, with significant load transfer across opposing crack faces being observed at minimum load (for R=0.1). These crack face tractions are associated with the plastic asperity created during overload. The traction forces holding the crack faces open at minimum load are, for the first time, used to directly quantify the associated stress intensity factor, Kmintract as a function of crack growth. While no crack shielding is expected, nor observed, for R=0.4, the variation in FCG rate after overload is explained by changes in effective R-ratio.